Consider an infinitely repeated partnership game between two players. In each period, the players select actions in the stage game, with common knowledge of the outcomes in all previous periods. In the stage game, player 1 selects a number x and player 2 selects a number y, where x and y are required to be in the interval from 0 to 10. Player 1's stage-game payoff is u1(x, y) = xy + 4x -x2
and player 2's stage-game payoff is
u2(x, y) = xy + 4y - y2
The players discount the future and have the same discount factor 8, which is strictly between 0 and 1.
1. Find the Nash equilibrium of the stage game.
2. Suppose that the players are interested in coordinating to select actions x = y = 10 in each period.